# Prerequisites for Differential Geometry

1. Dec 22, 2004

### hangman1414

Hello, I was wondering what you guys think is the absolute minimum requirements for learning Differential Geometry properly and also how would you go about learning it once you got to that point, recommended books, websites, etc. I am learning on my own because of some short circuit in my brain that makes me enjoy that sort of thing, lol. Thanks in advance, and nice board you guys have here!

2. Dec 22, 2004

### dextercioby

1).A very solid course on calculus.Starting from "0".From the basic elements of topology,up untill triple/multiple integrals and Gauss-Ostrogradsky formula.
2).A course on linear algebra would be useful.But not essential as 1),4)
3).A course on functional analysis including topology and the theory of measure and integration.Again not as important as 1),4)
4).A course (or at least very solid knowledge) of ordinary euclidean 3D&2D geometry.

About books on Differential Geometry in English i cannot give any reference,as i learnt the basics for phyiscs from online resources.There are many free online outlines and lecture notes on DiffGeom online,especially at the American Universities/Colleges.I believe you can searsh this forum for many links.

Daniel.

Edit:Check this thread and especially post no.12:

Last edited: Dec 22, 2004
3. Dec 22, 2004

### mathwonk

4. Dec 23, 2004

### pmb_phy

Note that the OP said "absolute minimum" requirements. Topology is not a absolute minimuim requirement to learn differential geometry. In fact I've never studied it myself and I know differential geometry.

A free online tensor calculus/diff geo text is at
http://www.math.odu.edu/~jhh/counter2.html

Pete

5. Dec 23, 2004

### dextercioby

:surprised :surprised :surprised :surprised :surprised :surprised :surprised :surprised

What's the basic notion of differential geometry????Is it "manifold"???Does the definition of a manifold assume the topologic structure on a set...???????
I would send you to the page 63 of the famous GR course online by Sean M.Carrol.I think you have it on your computer.If not,it has the address:
arXiv:gr-qc/9712019 v1 3 Dec 1997

Daniel.

PS.I hope u're not implying that Diff.Geom. can be thought without the concept of manifold.
That would be the differential geometry of Gauss,wouldn't it??Not the one of Cartan...
What kind of differential geometry do u know...????In whose boat would u be??In the one of Einstein,or in the one of Hermann Weyl and David Hilbert??

6. Dec 23, 2004

### dextercioby

I have the book that u reccomended sinc last year.It's pretty good.Basics,both in GR (calculative examples,brings example of Landau & Lifschitz),pretty thorough in fluid dynamics,has the only online treatment i found on Boltmzann equation treatment of fluids.Pretty weak with em.stuff.Worths reading.

Daniel.

7. Dec 23, 2004

### pmb_phy

Differential geometry is the study of Riemannian manifolds.
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects.

A precise knowledge of the later is not required for a minimum understanding to start reading Diff geo.
The OP does not need to go out and read a text on topology before he starts reading a text on diff geo.

Keep in mind what the OP asked for. He wanted to know what the absolute minimum requirements are. I didn't say that topology is unrelated to diff geo. This is equivalent to me saying that calculus is required, yet a detailed study of real analysis is not even though the later is the basis for the former.

Pete

Last edited: Dec 23, 2004
8. Dec 23, 2004

### dextercioby

I knew that differential geomatry is a study of MANIFOLDS,not of a subset on which u put a connection and a metric (Riemannian manifolds).Maybe we should ask a mathematician who knows better... :tongue2:

Who said a "precise" knowledge??But the definition of manifolds involves a few concepts of topology and differential calculus.

1.What is OP??
2.I advise him to do that.Let's see whose advice he follows... I guess it depends on how well he wants to learn...

Okay...absolute minimum requirements would mean reading Gauss' articles on geometry of curves and surfaces. :tongue2:

Daniel.

PS.U know that a hot branch of mathematics is differential topology...

9. Dec 23, 2004

### pmb_phy

The OP simple wants to know what he needs to get started learning diff geo.
All one needs to know is what a manifold is and a rigorous definition is not required for an intro to diff geo. If one thinks of a "manifold" simply as something which looks "locally" like a bit of n-dimensional Euclidean space Rn then you're all set.
Original Poster
You're missing the point. He wanted to know what he needed to start learning. He didn't imply that he'd stop there. Its good to get a start simply with calculus, linear diff eq, linear algebra and vector analysis. Those are the basics. With that under his belt he can start (not end) learning diff geo.

Pmb

10. Dec 23, 2004

### dextercioby

To make an analogy with a subject dear to me,it's like saying:
Q:"What is the basics/minimum for learning QM???????????"
A:"If you know that a Hilbert space is a vector space with a scalar product, <<then you're all set>>".

Daniel.

11. Dec 23, 2004

### pmb_phy

If I were to use that analogy then the question would be "What minimal math/physics do I need to know to start learning quantum theory". You don't [n]need[/i] to know Hilbert space.

You're giving him those things which are sufficient. He was not asking for that which he needs for a sufficient background. He's asking for the necessary background.

Pete

12. Dec 23, 2004

### quetzalcoatl9

wow

this is the exact kind of response that terrified me when i asked people who supposedly "were in the know". with age i have learned, in the end, that they knew very little.

my answer to the original poster: dont listen to anything anyone on this thread says (including myself). just try things your way.

of course you do not absolutely need to read a book on topology before picking up differential geometry. most differential geometry books will give you a rigorous definition of a differentiable manifold. that is all.

if you want more than that, great, get a topology book and read your heart out.

but there is no sense scaring a beginner about such things.

hangman, what i did was:

1) learned multivariable calculus
2) learned some very basic linear algebra
3) opened a book on differential geometry

as time progressed i learned many other things outside of the scope of that simple list. but if you do that, you will be on your path...

13. Dec 26, 2004

### mathwonk

I would like to point out one small thing that has apparently escaped some people posting advice here. The original question was not "what is the minimum foreknowledge recommended for starting to read differential geometry?" It was, what is the minimum needed to "learn Differential Geometry PROPERLY".

To understand what one is reading "properly", and put it into some context, requires more than simply to start reading. Of course the absolute minimum required to open a book on the subject is nothing but an ability to read the language the book is written in.

14. Dec 26, 2004

### Hurkyl

Staff Emeritus
One interesting approach (that I often take) is this:

Try to read differential geometry, and from that you can learn what other topics you should study.

15. Dec 26, 2004

### dextercioby

In my posts i virtually (wrongly) excluded the idea that he could find a decent book on diff.geom.which has an introductory chapter dealing with the basis of topology.If that kind of books exist,then he should be looking for them.

Daniel.

PS.To the OP:"Good luck!!".

Last edited: Dec 26, 2004
16. Dec 26, 2004

### mathwonk

some people have apparently learned differential geometry without knowing topology. I will add that from the opposite perspective I have studied topology and still do not feel I understand differential topology. It is hopeless to be fully prepared to understand anything. Learning is a process of going up and back and also around and around with a topic as it gradually becoems more clear. What one is satisifed as "understanding a subject" also evolves. With me I feel I understand a subject if I can see how every important idea was thought of and how every proof works, and have become independent of written expositions of the subject, since I see so clearly the workings that I never again need to refer to them. In particular I can give every proof, and even contrive new proofs, of every idea. I also wish to be able to explain everything, motivate everything, and illustrate everything in simple examples. Also one should be able to apply it and teach it. It takes me a long time to reach this level, and in fact I have not yet reached it with any subject, even calculus of one variable, although I have been studying and teaching it for over 40 years. indeen every year i learn something new about calculus as i teach it and try to explain it again. this year for instance I learned the proper hypotheses for the mean value theorem. i.e. how much should assume to have a constant function? continuity and derivative zero a.e. is not enough, but lipschitz continuity and derivative zero a.e. is enough. So lipschtiz continuity is a crucial property that an indefinite integral has, stronger than continuity. i.e. the indefinite integral of a riemann integrable function is a lipschitz continuous function that has a derivative almost everywhere. moreover within the realm of lipschitz continuous functions it is determined up to a constant by that derivative a.e.

I had read this theory in real variable courses in reference to lebesgue integration (the relevant concept there is "absolute continuity") but never understood it until this year. I have long referred to the radon nikodym theorem ignorantly as a typical example of a useless technical theorem in real analysis, and have argued for its omission from the phd syllabus. now after thoughtfully teaching one variable inhtegral calculus yet again, I finally understand its importance as characterizing which functions are integrals of other functions. so anytime anyone says you do not need to know such and such, consider who is saying this. maybe they themselves also do not understand the topic. at least this has often been true of me and my advice of this nature. all it means is I myself do not understand why something is needed or useful.

to make a long story longer, you will never regret any amount of topology you learn, as a backdrop to geometry or anything else. topology is the most basic aspect of geometry. but if your main interest is differentisal geometry, you may just plunge in learning it. however remember that your understanding will deepen as you learn more auxiliary subjects like topology and even algebra. Indeed tensors cannot be properly understood without some multilinear algebra.

Last edited: Dec 26, 2004
17. Dec 26, 2004

### hangman1414

Hello again, I thank all of you for your thoughtful replies. I think Mathwonk hit the nail on the head with his interpretation of my original post. By properly I meant I do not wish to take many shortcuts that may hinder my understanding of differential geometry, but I also do not want to learn anything that is not neccesary or related to the subject. A good understanding of differential geometry is my ultimate goal at this point, and if that takes learning alot of linear algebra (which I have a decent understanding of right now) then that is fine, if that's what it takes to realize my goal.
Also, the other reason I posted was I had tried a number of years ago to read a differential geometry book and was lost almost from the beginning, I didn't want that to happen again.
Thank you for your time, and take care!

18. Dec 26, 2004

### mathwonk

I will not necessarily claim that one must know differential manifolds to study differential geometry but it is entirely possible that it is necessary in order to understand the subject properly. michael spivak's treament spends the first volume on a detailed introduction to differential manifolds and then in the second volume, explains in equal detail the fundamental controbutions to differential geometry of Gauss and Riemann, including translations of their original papers. Of course the theory in spivak's volume one did not exist when gauss and riemann did their work, so it cannot have been necessary to it, but it is very helpful to us in understadning what they did. here are spivaks own remarks from his preface:

"For many years I have wanted to write the Great American Differential Geometry book. Today a dilemma confronts any one intent on penetrating the mysteries of differential geometry. On the one hand, one can consult numerous classical treatments of the subject in an attempt to form some idea how the concepts within it developed. Unfortunately, a modern mathematical education tends to make classical mathematical works inaccessible, particularly those in differential geometry. On the other hand, one can now find texts as modern in spirit, and as clean in exposition, as Bourbaki's Algebra. But a thorough study of these books usually leaves one unprepared to consult classical works, and entirely ignorant of the relationship between elegant modern constructions and their classical counterparts. ... no one denies that modern definitions are clear, elegant, and precise; it's just that it's impossible to comprehend how any one ever thought of them. And even after one does master a modern treatment of differential geometry, other modern treatments often appear simply to be about totally different subjects.

There are two main premises on which these notes are based. The first premise is that it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry.

The second premise for these notes is that in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature. The second volume of these notes gives a detailed exposition of the fundamental papers of Gauss and Riemann."

i myself suggest getting both books and reading volume two for the exciting geometry and referring back to volume one as one finds oneself wanting to understand volume two better. i.e. use vol 2 as motiovation to plow through vol 1.

19. Dec 27, 2004

### mathwonk

on reflection, since volumes 1 and 2 of spivaks differential geometry comprise over 1000 pages I guess, i suggest that the minimum background needed to understand differential geometry properly is a good grasp of several variables calculus, as explained wonderfully and succintly in spivaks calculus onmanifolds, about 140 pages.

20. Dec 28, 2004

### Perion

There's like five volumes in the total set of Spivaks A Comprehensive Introduction to Differential Geometry. See the five volume links at mathpop.com for volume contents and reviews. How would you like the totality of that information swimming about in your head :surprised

BTW - anyone thinking about Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus as some sort of introduction should read the reviews (especially Kishan Yerubandi's) at https://www.amazon.com/exec/obidos/tg/detail/-/0805390219/103-7004737-3250259?v=glance .

Perion

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